My best recommendation is to stop thinking about this character as a girl first and fighter second. You’re trying to come up with ways to make the fighting possible for her, instead of accepting that combat is a skill that can be developed by anyone given the proper amount of training and dedication. What weapon would you give this character if they were male?
That’s your answer.
As for picking weapons, I tend to pick weapons as a part of character creation and developing backstory (that blows up a little if the character is already established). I have a habit of doing this the same way I would write a crime: Motive. Method. Opportunity.
Motive: Why did this character want to learn to fight? What reason did they have to seek out training?
Most times, even in a family of established fighters, a character has to make the decision to train and to fight. This decision is a personal one and it can be anything from a desire for self protection to dreaming of being a knight in ballad. If you are working with a setting where female warriors are uncommon, then the character’s motivation for going against societal norms becomes that much more important.
Learning to fight is hard work and depending on that character’s background may well ruin any chance at conventional beauty/traditional womanhood/marriage opportunities that will better the standing of their families. It’s more than just an unusual choice, depending on the setting and gender constraints it could very well be an incredibly selfish one.
So, it’s important to establish that as part of the character.
Method: Who taught them? The good combatants have a teacher and the sword is a weapon that requires instruction, both in the manner of caring for the weapon and how to use it against other opponents. The character is going to need a teacher who can teach them to use that specific version of the weapon.
Did they have an in house tutor like Brienne of Tarth or Arya Starke? Did they receive their training when they joined the local military or militia? Did they have a parent train them? Were they carrying a blade that was common amongst peasants of their time like the arming sword or a weapon that was more regularly associated with the nobility like the long sword?
Opportunity: And what is a method if the character has no opportunity to take it? Think about your character’s background and social constraints, then pick a path that makes the most sense for them and was common for the people of their time (or the time/culture you’re basing it off of). The method they use will inevitably lead them to the right weapon.
This is where research is your friend, by narrowing down your path to profession and time period, you can better establish what your options are.
Remember: any weapon will work. Combat is a skill that can be learned and the only real physical barrier to entry is how hard you’re willing to work to learn it and the opportunities given to learn.
I didn’t pick taekwondo because it was the best suited to my size and body type, I picked the Ernie Reyes organization because they put on a performance at my elementary school that I really enjoyed. I saw it, said “I want to be able to do that”, took home the flier, and my parents signed me up.
I knew a lot of other kids (both boys and girls) who got into martial arts because they loved Ninja Turtles and Power Rangers.
The longbow versus crossbow question is actually fairly easy, both require a fair amount of strength to wield, but the truth is that care for the weapon is the most important point to maintain ease of drawing. Both require regular oiling and careful, specialized handling to ensure that they remain in a ready state of use.
The longbow is for characters like hunters, scouts, and nobles. Someone who grew up learning to or needing to hunt as a means for providing for their families. It can fire more rapidly than a crossbow, but requires more time to learn, more practice, and more training to be used effectively. In mass combat, archers were used in the same manner modern artillery is used today. The crossbow surpassed the longbow for the same reason that the gun surpassed the crossbow: it took a shorter amount of time to become as or more deadly than the other weapon, thus cheaper to replace when your troops fell. A lost archer is one to two to ten years of experience, compared to a lost crossbowman or gunman which is “point that way and fire”.
The crossbow is probably for a character who was trained via the military. A military trained character, depending on the time frame, will also be proficient in the use of anti cavalry tactics and pole arms. A female military conscript could easily just be a peasant girl whose mother dressed her up as a boy to either hide her from the men or hide a more valuable male sibling from the soldiers looking for recruits. It was not uncommon for peasants in the medieval period to be called up as levies to support their lord on the battlefield. They were usually just handed a spear and sent off to die, but there might be some workable ideas in there.
Training molds the body into a more suitable shape for the physical activity. So, if your fighter is a noblewoman, don’t expect her to keep the secret for long. Also, servants talk. People are observant. They will know.
Some things to think about.